3.4 Interpretive Guide

The following table provides a guide for interpreting your score, on the basis of the number of questions you got right.

Remember, you should not compare the number of questions you got right in each section. Instead, you should compare how your response rated in each section.

3.5 Quantitative Answer Explanations

Problem Solving

The following discussion is intended to familiarize you with the most efficient and effective approaches to the kinds of problems common to problem solving questions. The particular questions in this chapter are generally representative of the kinds of quantitative questions you will encounter on the GMAT exam. Remember that it is the problem solving strategy that is important, not the specific details of a particular question.

1.Last month a certain music club offered a discount to preferred customers. After the first compact disc purchased, preferred customers paid $3.99 for each additional compact disc purchased. If a preferred customer purchased a total of 6 compact discs and paid $15.95 for the first compact disc, then the dollar amount that the customer paid for the 6 compact discs is equivalent to which of the following?

(A)

(B)

(C)

(D)

(E)

Arithmetic Operations on rational numbers

The cost of the 6 compact discs, with $15.95 for the first one and $3.99 for the other 5 discs, can be expressed as . It is clear from looking at the answer choices that some regrouping of the values is needed because none of the answer choices uses $3.99 in the calculation.

If $4.00 is used instead of $3.99, each one of the 5 additional compact discs is calculated at $0.01 too much, and the total cost is too high. There is an overage of $0.05 that must be subtracted from the $15.95, or thus . Therefore, the cost can be expressed as 5(4.00) + 15.90.

The correct answer is A.

2.The average (arithmetic mean) of the integers from 200 to 400, inclusive, is how much greater than the average of the integers from 50 to 100, inclusive?

(A)150

(B)175

(C)200

(D)225

(E) 300 Arithmetic Statistics

In the list of integers from 200 to 400 inclusive, the middle value is 300. For every integer above 300, there exists an integer below 300 that is the same distance away from 300; thus the average of the integers from 200 to 400, inclusive, will be kept at 300. In the same manner, the average of the integers from 50 to 100, inclusive, is 75.

The difference is .

The correct answer is D.

3.The sequence a1, a2, a3, . . . ,an, . . . is such that for all . If and , what is the value of a6?

(A)12

(B)16

(C)20

(D)24

(E)28

Algebra Applied problems

According to this formula, it is necessary to know the two prior terms in the sequence to determine the value of a term; that is, it is necessary to know both and to find an. Therefore, to find a6, the values of a5 and a4 must be determined.

To find a4, let , which makes and . Then, by substituting the given values into the formula

substitute known values

multiply both sides subtract 4 from both sides

Then, letting , substitute the known values:

substitute known values simplify

The correct answer is E.

4.Among a group of 2,500 people, 35 percent invest in municipal bonds, 18 percent invest in oil stocks, and 7 percent invest in both municipal bonds and oil stocks. If 1 person is to be randomly selected from the 2,500 people, what is the probability that the person selected will be one who invests in municipal bonds but NOT in oil stocks?

(A)

(B)

(C)

(D)

(E)

Arithmetic Probability

Since there are 2,500 people, 875 people invest in municipal bonds, and

of those people invest in both municipal bonds and oil stocks. Therefore, there are people who invest in municipal bonds but not in oil stocks. Probability of an event .

Probability of investing in municipal bonds but not in oil stocks .

The correct answer is B.

5.A closed cylindrical tank contains cubic feet of water and is filled to half its capacity. When the tank is placed upright on its circular base on level ground, the height of the water in the tank is 4 feet. When the tank is placed on its side on level ground, what is the height, in feet, of the surface of the water above the ground?

(A)2

(B)3

(C)4

(D)6

(E)9

Geometry Volume

Since the cylinder is half full, it will be filled to half its height, whether it is upright or on its side. When the cylinder is on its side, half its height is equal to its radius.

Using the information about the volume of water in the upright cylinder, solve for this radius to determine the height of the water when the cylinder is on its side.

known volume of water is

substitute 4 for h; divide both sides by

solve for r

radius height of the water in the cylinder on its side

The correct answer is B.

6.A marketing firm determined that, of 200 households surveyed, 80 used neither Brand A nor Brand B soap, 60 used only Brand A soap, and for every household that used both brands of soap, 3 used only Brand B soap. How many of the 200 households surveyed used both brands of soap?

(A)15

(B)20

(C)30

(D)40

(E)45

Arithmetic Operations on rational numbers

Since it is given that 80 households use neither Brand A nor Brand B, then must use Brand A, Brand B, or both. It is also given that 60 households use only

Brand A and that three times as many households use Brand B exclusively as use both brands. If x is the number of households that use both Brand A and Brand B, then 3x use Brand B alone. A Venn diagram can be helpful for visualizing the logic of the given information for this item:

All the sections in the circles can be added up and set equal to 120, and then the equation can be solved for x:

combine like terms

subtract 60 from both sides

divide both sides by 4

The correct answer is A.

7.A certain club has 10 members, including Harry. One of the 10 members is to be chosen at random to be the president, one of the remaining 9 members is to be chosen at random to be the secretary, and one of the remaining 8 members is to be

chosen at random to be the treasurer. What is the probability that Harry will be either the member chosen to be the secretary or the member chosen to be the treasurer?

(A)

(B)

(C)

(D)

(E)

Arithmetic Probability

Two probabilities must be calculated here: (1) the probability of Harry’s being chosen for secretary and (2) the probability of Harry’s being chosen for treasurer. For any probability, the probability of an event’s occurring .

(1) If Harry is to be secretary, he first CANNOT have been chosen for president, and then he must be chosen for secretary. The probability that he will be chosen for president is , so the probability of his NOT being chosen for president is . Then, the probability of his being chosen for secretary is . Once he is chosen, the probability that he will be selected for treasurer is 0, so the probability that he will NOT be selected for treasurer is .

Thus, the probability that Harry will be chosen for secretary is .

(2) If Harry is to be treasurer, he needs to be NOT chosen for president, then NOT chosen for secretary, and then finally chosen for treasurer. The probability that he will NOT be chosen for president is again . The probability of his NOT being chosen for secretary is . The probability of his being chosen for treasurer is , so the probability that Harry will be

chosen for treasurer is .

(3) So, finally, the probability of Harry’s being chosen as either secretary or treasurer is thus .

The correct answer is E.

8.If a certain toy store’s revenue in November was of its revenue in December and its revenue in January was of its revenue in November, then the store’s revenue in December was how many times the average (arithmetic mean) of its revenues in November and January?

(A)

(B)

(C)

(D)2

(E) 4

Arithmetic Statistics

Let n be the store’s revenue in November, d be the store’s revenue in December, and j be the store’s revenue in January. The information from the problem can be expressed as and Substituting for n in the second equation gives

. Then, the average of the revenues in November and January can be found by using these values in the formula

, as follows:

Solve for the store’s revenue in December by multiplying both sides of this equation by 4:

average

Thus, the store’s revenue in December was 4 times its average revenue in November and January.

The correct answer is E.

9.A researcher computed the mean, the median, and the standard deviation for a set of performance scores. If 5 were to be added to each score, which of these three statistics would change?

(A)The mean only

(B)The median only

(C)The standard deviation only

(D)The mean and the median

(E)The mean and the standard deviation

Arithmetic Statistics

If 5 were added to each score, the mean would go up by 5, as would the median. However, the spread of the values would remain the same, simply centered around a new value. So, the standard deviation would NOT change.

The correct answer is D.

10.In the figure shown, what is the value of ?

(A)45

(B)90

(C)180

(D)270

(E)360

Geometry Angles and their measure

In the following figure, the center section of the star is a pentagon.

The sum of the interior angles of any polygon is , where n is the number of sides. Thus, .

Each of the interior angles of the pentagon defines a triangle with two of the angles at the points of the star. This gives the following five equations:

Summing these 5 equations gives:

.

Substituting 540 for gives:

.

From this:

subtract 540 from both sides

factor out 2 on the left side

divide both sides by 2

The correct answer is C.

11.Of the three-digit integers greater than 700, how many have two digits that are equal to each other and the remaining digit different from the other two?

(A)90

(B)82

(C)80

(D)45

(E)36

Arithmetic Properties of numbers

In three-digit integers, there are three pairs of digits that can be the same while the other digit is different: tens and ones, hundreds and tens, and hundreds and ones. In each of these pairs, there are 9 options for having the third digit be different from the other two. The single exception to this is in the 700–799 set, where the number 700 cannot be included because the problem calls for integers “greater than 700”. So, in the 700–799 set, there are only 8 options for when the tens and ones are the same. This is shown in the table below.

Number of digits available for the third digit when two given digits are the same

Thus, of the three-digit integers greater than 700, there are numbers that have two digits that are equal to each other when the remaining digit is different from these two.

The correct answer is C.

12.Positive integer y is 50 percent of 50 percent of positive integer x, and y percent of x equals 100. What is the value of x?

(A)50

(B)100

(C)200

(D)1,000

(E)2,000

Arithmetic; Algebra Percents; Simultaneous equations

Because y is a positive integer, y percent is notated as . According to the problem,

and .

The first equation simplifies to , and multiplying the second equation by 100 gives .

Substituting the simplified first equation into this second equation gives:

simplify left side

divide both sides by 0.25

solve for the value of x

The correct answer is C.

13.If s and t are positive integers such that , which of the following could be the remainder when s is divided by t?

(A)2

(B)4

(C)8

(D)20

(E)45

Arithmetic Operations on rational numbers

By using a long division model, it can be seen that the remainder after dividing s by t is :

Then, the given equation can be written as . By splitting portions of t into its integer multiple and its decimal multiple, this becomes , or , which is the remainder. So, . Test the answer choices to find the situation in which t is an integer.

The correct answer is E.

14.Of the 84 parents who attended a meeting at a school, 35 volunteered to supervise children during the school picnic and 11 volunteered both to supervise children during the picnic and to bring refreshments to the picnic. If the number of parents who volunteered to bring refreshments was 1.5 times the number of parents who neither volunteered to supervise children during the picnic nor volunteered to bring refreshments, how many of the parents volunteered to bring refreshments?

(A)25

(B)36

(C)38

(D)42

(E)45

Arithmetic Operations on rational numbers

Out of the 35 parents who agreed to supervise children during the school picnic, 11 parents are also bringing refreshments, so 24 parents are only supervising children. Let x be the number of parents who volunteered to bring refreshments, and let y be the number of parents who declined to supervise or to bring refreshments. The fact that the number of parents who volunteered to bring refreshments is 1.5 times the number who did not volunteer at all can then be expressed as . A Venn diagram, such as the one below, can be helpful in answering problems of this kind.

Then, the sum of the sections can be set equal to the total number of parents at the picnic, and the equation can be solved for y:

sum of sections total parents at picnic

subtract 24 from each side

subtract x from each side Then, substituting the value for y in the equation x = 1.5y gives the following:

distribute the 1.5

add 1.5x to both sides divide both sides by 2.5

The correct answer is B.

15.The product of all the prime numbers less than 20 is closest to which of the following powers of 10?

(A)109

(B)108

(C)107

(D)106

(E)105

Arithmetic Properties of numbers

The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Their product is 9,699,690 (arrived at as follows:

.

This is closest to .

The correct answer is C.

16.If then,

(A)1

(B)4

(C)

(D)

(E)

Algebra Second-degree equations

Work with the equation to create 4x2 on one side.

square both sides

move all non-square-root terms to one side (i.e., subtract 2x and 1) divide both sides by 2

square both sides

isolate the 4x2 (add 4x and subtract 1 from both sides)

The correct answer is E.

17.If , what is the value of ?

(A)

(B)

(C)

(D)

(E)

Arithmetic Operations on radical expressions

Work the problem.

Since , then .

The correct answer is D.

18.If n is the product of the integers from 1 to 8, inclusive, how many different prime factors greater than 1 does n have?

(A)Four

(B)Five

(C)Six

(D)Seven

(E)Eight

Arithmetic Properties of numbers

If n is the product of the integers from 1 to 8, then its prime factors will be the prime numbers from 1 to 8. There are four prime numbers between 1 and 8: 2, 3, 5, and 7.

The correct answer is A.

19.If k is an integer and , for how many different values of k is there a triangle with sides of lengths 2, 7, and k?

(A)One

(B)Two

(C)Three

(D)Four

(E)Five

Geometry Triangles

In a triangle, the sum of the smaller two sides must be larger than the largest side.

For k values 3, 4, 5, and 6, the only triangle possible is 2, 7, and because only . For k values 3, 4, and 5, the sum of the smaller two sides is not larger than the third side; thus, 6 is the only possible value of k that satisfies the conditions.

The correct answer is A.

20.A right circular cone is inscribed in a hemisphere so that the base of the cone coincides with the base of the hemisphere. What is the ratio of the height of the cone to the radius of the hemisphere?

(A)

(B)1:1

(C)

(D)

(E)2:1

Geometry Volume

As the diagram below shows, the height of the cone will be the radius of the hemisphere, so the ratio is 1:1.

The correct answer is B.

21.John deposited $10,000 to open a new savings account that earned 4 percent annual interest, compounded quarterly. If there were no other transactions in the account, what was the amount of money in John’s account 6 months after the account was opened?

(A)$10,100

(B)$10,101

(C)$10,200

(D)$10,201

(E)$10,400

Arithmetic Operations on rational numbers

Since John’s account is compounded quarterly, he receives of his annual interest,

or 1%, every 3 months. This is added to the amount already in the account to accrue interest for the next quarter. After 6 months, this process will have occurred twice, so the amount in John’s account will then be

The correct answer is D.

22.A container in the shape of a right circular cylinder is full of water. If the volume of water in the container is 36 cubic inches and the height of the container is 9 inches, what is the diameter of the base of the cylinder, in inches?

(A)

(B)

(C)

(D)

(E)

Geometry Volume

For a right cylinder, volume (radius)2(height). Since the volume of water is 36 cubic inches and since this represents the container, the water is occupying the container’s height, or inches. Let r be the radius of the cylinder.

divide both sides by 4.5

divide both sides by

take the square root of both sides simplify the to get the radius

Then, since the diameter is twice the length of the radius, the diameter equals .

The correct answer is E.

23.If the positive integer x is a multiple of 4 and the positive integer y is a multiple of 6, then xy must be a multiple of which of the following?

I. 8

II. 12

III. 18

(A)II only

(B)I and II only

(C)I and III only

(D)II and III only

(E)I, II, and III

Arithmetic Properties of numbers

The product xy must be a multiple of and any of its factors. Test each alternative.

I. 8 is a factor of 24 MUST be a multiple of 8 II. 12 is a factor of 24 MUST be a multiple of 12

III. 18 is NOT a factor of 24 NEED NOT be a multiple of 18

The correct answer is B.

24.Aaron will jog from home at x miles per hour and then walk back home by the same route at y miles per hour. How many miles from home can Aaron jog so that he spends a total of t hours jogging and walking?

(A)

(B)

(C)

(D)

(E)

Algebra Simplifying algebraic expressions

Let j be the number of hours Aaron spends jogging; then let be the total number of hours he spends walking. It can be stated that Aaron jogs a distance of xj miles and walks a distance of miles. Because Aaron travels the same route, the miles jogged must equal the miles walked, and they can be set equal.

set number of miles equal to each other

distribute the y

add jy to both sides to get all terms with j to one side

factor out the j

divide both sides by So, the number of hours Aaron spends jogging is .

The number of miles he can jog is xj or, by substitution of this value of j, .

The correct answer is C.

are the only factors of p.

The correct answer is E; both statements together are still not sufficient.

27.If the length of Wanda’s telephone call was rounded up to the nearest whole minute by her telephone company, then Wanda was charged for how many minutes for her telephone call?

(1)The total charge for Wanda’s telephone call was $6.50.

(2)Wanda was charged $0.50 more for the first minute of the telephone call than for each minute after the first.

Arithmetic Arithmetic operations

(1)This does not give any information as to the call’s cost per minute; NOT sufficient.

(2)From this, it can be determined only that the call was longer than one minute and that the charge for the first minute was $0.50 more than the charge for each succeeding minute; NOT sufficient.

Taking (1) and (2) together, the number of minutes cannot be determined as long as the cost of each minute after the first is unknown. For example, if the cost of each minute after the first minute were $0.40, then the cost of the first minute would be $0.90. Then the total cost of the other minutes would be , and would yield 14. In this case, the time of the call would be minutes. If, however, the cost of each minute after the first minute were $0.15, then the cost of the first minute would be $0.65. Then would be $5.85, and this in turn, when divided by $0.15, would yield 39 minutes, for a total call length of 40 minutes. More information on the cost of each minute after the first minute is still needed.

The correct answer is E; both statements together are still not sufficient.

28.What is the perimeter of isosceles triangle MNP?

(1)

(2)

Geometry Triangles

The perimeter of a triangle is the sum of all three sides. In the case of an isosceles triangle, two of the sides are equal. To determine the perimeter of this triangle, it is necessary to know both the length of an equal side and the length of the base of the triangle.

(1)Only gives the length of one side; NOT sufficient.

(2)Only gives the length of one side; NOT sufficient.

Since it is unclear whether MN or NP is one of the equal sides, it is not possible to

determine the length of the third side or the perimeter of the triangle. The perimeter could be either or .

The correct answer is E; both statements together are still not sufficient.

29.In a survey of retailers, what percent had purchased computers for business purposes?

(1)85 percent of the retailers surveyed who owned their own store had purchased computers for business purposes.

(2)40 percent of the retailers surveyed owned their own store.

Arithmetic Percents

(1)With only this, it cannot be known what percent of the retailers not owning their own store had purchased computers, and so it cannot be known how many retailers purchased computers overall; NOT sufficient.

(2)While this permits the percent of owners and nonowners in the survey to be deduced, the overall percent of retailers who had purchased computers cannot be determined; NOT sufficient.

Using the information from both (1) and (2), the percent of surveyed ownerretailers who had purchased computers can be deduced, and the percent of nonowner-retailers can also be deduced. However, the information that would permit a determination of either the percent of nonowner-retailers who had purchased computers or the overall percent of all retailers (both owners and nonowners) who had purchased computers is still not provided.

The correct answer is E; both statements together are still not sufficient.

30.The only gift certificates that a certain store sold yesterday were worth either $100 each or $10 each. If the store sold a total of 20 gift certificates yesterday, how many gift certificates worth $10 each did the store sell yesterday?

(1)The gift certificates sold by the store yesterday were worth a total of between $1,650 and $1,800.

(2)Yesterday the store sold more than 15 gift certificates worth $100 each.

Algebra Applied problems; Simultaneous equations; Inequalities

Let x represent the number of $100 certificates sold, and let y represent the number of $10 certificates sold. Then the given information can be expressed as or thus . The value of the $100 certificates sold is 100x, and the value of the $10 certificates sold is 10y.

(1) From this, it is known that 1,650. Since , this value can be substituted for y, and the inequality can be solved for x:

substitute for y distribute

simplify

subtract 200 from both sides

Thus, more than 16 of the $100 certificates were sold. If 17 $100 certificates were sold, then it must be that 3 $10 certificates were also sold for a total of $1,730, which satisfies the condition of being between $1,650 and $1,800. If, however, 18 $100 certificates were sold, then it must be that 2 $10 certificates were sold, and this totals $1,820, which is more than $1,800 and fails to satisfy the condition. Therefore, 3 of the $10 certificates were sold; SUFFICIENT.

(2) From this it can be known only that the number of $10 certificates sold was 4 or fewer; NOT sufficient.

The correct answer is A; statement 1 alone is sufficient.

31.Is the standard deviation of the set of measurements x1, x2, x3, x4, . . . , x20 less than 3?

(1)The variance for the set of measurements is 4.

(2)For each measurement, the difference between the mean and that measurement is 2.

Arithmetic Statistics

In determining the standard deviation, the difference between each measurement and the mean is squared, and then the squared differences are added and divided by the number of measurements. The quotient is the variance and the positive square root of the variance is the standard deviation.

(1)If the variance is 4, then the standard deviation , which is less than 3; SUFFICIENT.

(2)For each measurement, the difference between the mean and that measurement is 2. Therefore, the square of each difference is 4, and the sum

of all the squares is . The standard deviation is , which is less than 3; SUFFICIENT.

The correct answer is D; each statement alone is sufficient.

32. Is the range of the integers 6, 3, y, 4, 5, and x greater than 9?

(1)

(2)

Arithmetic Statistics

The range of a set of integers is equal to the difference between the largest integer

and the smallest integer. The range of the set of integers 3, 4, 5, and 6 is 3, which is derived from .

(1) Although it is known that , the value of x is unknown. If, for example,

, then the value of y would be greater than 3. However, if , then the value of y would be greater than 6, and, since 6 would no longer be the largest integer, the range would be affected. Because the actual values of x and y are unknown, the value of the range is also unknown; NOT sufficient.

(2) If and , then x could be 4 and y could be 5. Then the range of the 6 integers would still be or 3. However, if x were 4 and y were 15, then the range of the 6 integers would be , or 12. There is no means to establish the values of x and y, beyond the fact that they both are greater than 3; NOT sufficient.

Taking (1) and (2) together, it is known that and that . Since the smallest integer that x could be is thus 4, then or . Therefore, the integer y must be 13 or larger. When y is equal to 13, the range of the 6 integers is , which is larger than 9. As y increases in value, the value of the range will also increase.

The correct answer is C; both statements together are sufficient.

33.Is ?

(1)

(2)

Algebra Inequalities

Note that .

(1)If , then since ; SUFFICIENT.

(2)If , then

add 2 to both sides

because implies

divide both sides by ;

SUFFICIENT.

The correct answer is D; each statement alone is sufficient.

34.Of the companies surveyed about the skills they required in prospective employees, 20 percent required both computer skills and writing skills. What percent of the companies surveyed required neither computer skills nor writing skills?

(1)Of those companies surveyed that required computer skills, half required writing skills.

(2) 45 percent of the companies surveyed required writing skills but not computer skills.

Arithmetic Percents

The surveyed companies could be placed into one of the following four categories:

1.Requiring computer skills and requiring writing skills

2.Requiring computer skills but not requiring writing skills

3.Not requiring computer skills but requiring writing skills

4.Not requiring either computer skills or writing skills

It is given that 20 percent of the surveyed companies fell into category 1. It is necessary to determine what percent of the surveyed companies fell into category 4.

(1) This helps identify the percentage in category 2. Since the companies that required computer skills also required writing skills (i.e., those in category 1), then the other of the companies that required computer skills did not require writing skills (thus category 2 category 1). However, this information only establishes that 20 percent required computer skills, but not writing skills; NOT sufficient.

(2) While this establishes category 3, that is, that 45 percent required writing skills but not computer skills, no further information is available; NOT sufficient.

Taking (1) and (2) together, the first three categories add up to 85 percent . Therefore, category 4 would be equal to 15 percent of the surveyed companies required neither computer skills nor writing skills.

The correct answer is C; both statements together are sufficient.

35.What is the value of ?

(1)

(2)

Algebra Firstand second-degree equations

(1)If 3q is added to both sides of this equation, it can be rewritten as . When each term is then divided by 3, it yields ; SUFFICIENT.

(2)When each term in this equation is divided by 5, it becomes ; SUFFICIENT.

The correct answer is D; each statement alone is sufficient.

36.If X and Y are points in a plane and X lies inside the circle C with center O and radius 2, does Y lie inside circle C?

(1)The length of line segment XY is 3.

(2)The length of line segment OY is 1.5.

Geometry Circles

(1)The maximum distance between two points that lie on a circle is equal to the diameter, or 2 times the radius. Since the radius of circle C is 2, the diameter in this case is 4. It cannot be assumed, however, that X and Y are points on the diameter; X can lie anywhere within the circle. When the distance between X and Y is 3, it is still possible either that Y is within the circle or that Y is outside the circle; NOT sufficient.

(2)If the length of the line segment OY is 1.5 and the circle has a radius of 2, then the distance from the center O to point Y is less than the radius, and point Y must therefore lie within the circle; SUFFICIENT.

The correct answer is B; statement 2 alone is sufficient.

37. Is ?

(1)

(2)

Algebra Firstand second-degree equations

(1)Since 2 has to be added to y in order to make it equal to x, it can be reasoned that ; SUFFICIENT.

(2)Multiplying both sides of this equation by 2 results in or . If y were 0, then x would be −2, and y would be greater than x. If y were a negative number like −2, then , and again y would be greater than x. However, if y were a positive number such as 4, then , and . Since there is no other information concerning the value of y, it cannot be determined if ; NOT sufficient.

The correct answer is A; statement 1 alone is sufficient.

38.If Paula drove the distance from her home to her college at an average speed that was greater than 70 kilometers per hour, did it take her less than 3 hours to drive this distance?

(1)The distance that Paula drove from her home to her college was greater than 200 kilometers.

(2)The distance that Paula drove from her home to her college was less than 205 kilometers.

Arithmetic Distance problem

A distance problem uses the formula distance rate time. To find the time, the formula would be rearranged as . To solve this

problem, it is necessary to know the rate (given here as 70 kilometers per hour) and the distance.

(1)If D is the distance Paula drove then and so and t may or may not be less than 3; NOT sufficient.

(2)If D is the distance Paula drove then and so ; SUFFICIENT.

The correct answer is B; statement 2 alone is sufficient.

39.In the xy-plane, if line k has negative slope and passes through the point (−5,r), is the x-intercept of line k positive?

(1)The slope of line k is −5.

(2)

Geometry Coordinate geometry

The x-intercept is the x-coordinate of the point in which the line k crosses the x- axis and would have the coordinates (x,0).

(1)Knowing the slope of the line does not help in determining the x- intercept, since from point (–5,r) the line k extends in both directions. Without knowing the value of r, the x-intercept could be −5 if r were 0, or it could be other numbers, both positive and negative, depending on the value of r; NOT sufficient.

(2)Knowing that suggests that the x-intercept is not −5; the point (–5,r), where r is a positive number, does lie in quadrant II. It could, however, be any point with an x-coordinate of −5 in that quadrant and line k could have any negative slope, and so the line k would vary with the value of r. Therefore, the x-intercept of line k cannot be determined; NOT sufficient.

Using (1) and (2) together does not help in the determination of the x-intercept, since the point (–5,r) could have any positive y-coordinate and thus line k could cross the x-axis at many different places.

The correct answer is E; both statements together are still not sufficient.

40.If $5,000 invested for one year at p percent simple annual interest yields $500, what amount must be invested at k percent simple annual interest for one year to yield the same number of dollars?

(1)

(2)

Arithmetic Interest problem

With simple annual interest, the formula to use is interest principal rate time. It is given that (year), so percent interest.

(1) If p is 10 percent, then is 0.08. Using the same formula, the time is again 1 year; the interest is the same amount; and the rate is 0.08, or 8 percent. Thus, , or principal $6,250; SUFFICIENT.

(2) If , then the rate is 8 percent, and the same formula and procedure as above are employed again; SUFFICIENT.

The correct answer is D; each statement alone is sufficient.

41.If , is ?

(1)

(2)

Algebra Inequalities

If , then either one of two cases holds true. Either and , or and

. In other words, in order for the term to be greater than zero, it must be true that either 1) both the numerator and denominator are greater than 0 or 2) both the numerator and denominator are less than 0.

(1)Regardless of whether is positive or negative, the positive or negative value of z must be in agreement with the sign of in order for . However, there is no information about z here; NOT sufficient.

(2)If , then must be less than 0. However, this statement gives no information about ; NOT sufficient.

This can be solved using (1) and (2) together. From (2), it is known that , and, going back to the original analysis, for the term to be greater than zero, must also be less than 0. If then . But from (1) so

.

The correct answer is C; both statements together are sufficient.

42.Does the integer k have at least three different positive prime factors?

(1) is an integer.

(2) is an integer.

Arithmetic Properties of numbers

(1)The prime factors of 15 are 3 and 5. So in this case, k has at least 2 different positive prime factors, but it is unknown if there are more positive prime factors; NOT sufficient.

(2)The prime factors of 10 are 2 and 5, showing that k has at least these 2

one given month. Does at least one person in the auditorium have a birthday in January?

(1)More of the people in the auditorium have birthdays in February than in March.

(2)Five of the people in the auditorium have birthdays in March.

Algebra Sets and functions

Because it is given that 6 is the greatest number of individuals who can have birthdays in any particular month, these 66 people could be evenly distributed across 11 of the 12 months of the year. That is to say, it could be possible for the distribution to be , and thus any given month, such as January, would not have a person with a birthday. Assume that January has no people with birthdays, and see if this assumption is disproved.

(1)The information that more people have February birthdays than March birthdays indicates that the distribution is not even. Therefore, March is underrepresented and must thus have fewer than 6 birthdays. Since no month can have more than 6 people with birthdays, and every month but January already has as many people with birthdays as it can have, January has to have at least 1 person with a birthday; SUFFICIENT.

(2)Again, March is underrepresented with only 5 birthdays, and none of the other months can have more than 6 birthdays. Therefore, the extra birthday (from March) must occur in January; SUFFICIENT.

The correct answer is D; each statement alone is sufficient.

46.Last year the average (arithmetic mean) salary of the 10 employees of Company X was $42,800. What is the average salary of the same 10 employees this year?

(1)For 8 of the 10 employees, this year’s salary is 15 percent greater than last year’s salary.

(2)For 2 of the 10 employees, this year’s salary is the same as last year’s salary.

Arithmetic Statistics

(1)Since all 10 employees did not receive the same 15 percent increase, it cannot be assumed that the mean this year is 15 percent higher than last year. It remains unknown whether these 8 salaries were the top 8 salaries, the bottom 8 salaries, or somewhere in-between. Without this type of information from last year, the mean for this year cannot be determined; NOT sufficient.

(2)If 2 salaries remained the same as last year, then 8 salaries changed. Without further information about the changes, the mean for this year cannot be determined; NOT sufficient.

Even taking (1) and (2) together, it remains impossible to tell the mean salary for this year without additional data.

The correct answer is E; both statements together are still not sufficient.

47.In a certain classroom, there are 80 books, of which 24 are fiction and 23 are written in Spanish. How many of the fiction books are written in Spanish?

(1)Of the fiction books, there are 6 more that are not written in Spanish than are written in Spanish.

(2)Of the books written in Spanish, there are 5 more nonfiction books than fiction books.

Algebra Sets and functions

Let x represent the fiction books that are written in Spanish. A table could be set up like the one below, filling in the information that is known or able to be known:

(1)If x represents the fiction books written in Spanish, then can now be used to represent the fiction books that are not written in Spanish. From the table above, it can be seen then that , or . Therefore, x, or the number of fiction books written in Spanish, is 9; SUFFICIENT.

(2)If x represents the fiction books written in Spanish, then can now be used to represent the nonfiction books written in Spanish. From the table, it can be said that , or . Therefore, x, or the number of fiction books written in Spanish, is 9; SUFFICIENT.

The correct answer is D; each statement alone is sufficient.

48.If p is the perimeter of rectangle Q, what is the value of p?

(1)Each diagonal of rectangle Q has length 10.

(2)The area of rectangle Q is 48.

Geometry Rectangles; Perimeter; Area

The perimeter of a rectangle is equal to 2 times the rectangle’s length plus 2 times the rectangle’s width, or . The diagonals of a rectangle are equal. In a rectangle, because a diagonal forms a right triangle, the length of a diagonal is equal to the square root of the length squared plus the width squared, or .

(1) If a diagonal , then , or, by squaring both sides, . Without knowing the value or the relationship between the other two sides of the

right triangle, it is impossible to solve for l or w, and thus for the perimeter of the rectangle; NOT sufficient.

(2) If the area of the rectangle is 48, then it can be stated that . However, without further information, the perimeter cannot be determined. For example, l could be 6 and w could be 8, and the perimeter would then be . However, it could also be that l is 4 and w is 12, and in that case the perimeter would be 32; NOT sufficient.

Using (1) and (2) together, it is possible to solve this problem. Since from (2) , then . Substituting this into from (1) the equation can be solved as follows:

substitution

multiply both sides by l2

move all terms to one side

factor like a quadratic

solve for l2

Since l is a length, it must be positive, so l is either 8 or 6. When , , and when , , both of which give the same perimeter.

The correct answer is C; both statements together are sufficient.